Spectral moments and linear models used for photoacoustic detection of crude oil in produced water

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University of Oslo

Department of Informatics

Spectral moments and linear models used for photo- acoustic detection of crude oil in

produced water

Fredrik Vogel

26th April 2001

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List of Tables

3.1 Test parameters. . . 9 5.1 The parameters in equation 5.1, their Standard deviation

(Sd.) and thep-value. . . . 27 5.2 The parameters in equation 5.2, their standard deviation

(Sd.) and thep-value. . . . 27 5.3 The four parameters from equation 5.3, their standard de-

viation (Sd.) and thep-value. . . . 29 5.4The parameters, from equation 5.4, their standard devi-

ation (Sd.) and thep-value. . . . 29 6.1 Parameter values from equations 6.9, 6.10 and 6.11. . . 50 7.1 The seven predictors of equation 7.1 is displayed with their

parametric value, standard deviation (Sd.), t-value and p-value 53 7.2 Cp-values after stepwise removal of variables in model 7.1

with corresponding Residual Sum of Squares. . . 54 7.3 Cp-values produced by diﬀerent models. . . . 57

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List of Figures

1.1 Correlation between discharge of oil and produced water . 3 2.1 Principle of Photoacoustic propagation . . . 6 3.1 Front view of the head unit. . . 8 3.2 OIWM installed in the water rig at the Hydro’s Research

Center in Porsgrunn. . . 10 3.3 Plot of the IR-method versus the dispenser oil concentration. 10 3.4 Typical photoacoustic signal from the OIWM. . . 12 5.1 All data, using oil concentration from the oil dispenser. . . 22 5.2 All data, using oil concentration from the IR-method. . . 23 5.3 Data showing photoacoustic response versus temperature

with corresponding regression lines. . . 25 5.4Data showing photoacoustic response versus IR-method oil

concentration with corresponding regression lines. . . 26 5.5 Plot of regression ﬁt of the oil concentration from the oil

dispenser. . . 28 5.6 Plot of regression ﬁt of the oil concentration from the IR-

method. . . 28 5.7 Plot of the residual error of the diﬀerent oil concentration

from the oil dispenser, found with equation 5.3. . . 30 5.8 Plot of the residual error of the diﬀerent oil concentration

from the IR-method, found with equation 5.4. . . 31 5.9 Plot of ﬁtted versus the monitored salinity values, found

with equation 5.5. . . 32 6.1 Plot of the Fourier transform of a normal OIWM signal. . . . 36 6.2 Order 500 FIR filter with linear phase. . . 37 6.3 ChebyshevII IIR filter of Order 3 with non linear phase. . . 38 6.4Plot of original, IIR filtered and FIR filtered signal. . . 39 6.5 Relation between the peak-to-peak value of the FIR filtered

signal and the temperature. . . 4 0 6.6 Relation between the peak-to-peak value plotted against tem-

perature. . . 4 0 vii

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viii LIST OF FIGURES

6.7 Visualization of the three ﬁrst spectral moments. . . 41

6.8 Plot of arithmetic mean. . . 4 3 6.9 Plot of geometric mean. . . 4 3 6.10 Properties of the lowpass filter. . . 4 5 6.11 The impulse response of the highpass filter. . . 4 6 6.12 Properties of the highpass filter. . . 4 7 6.13 The decimated signal, the filtered in blue and the original in red. . . 4 7 6.14Output from using the arithmetic mean on the decimated and filtered signal. . . 4 9 7.1 Correlation plot between some of the key variables. . . 55

7.2 The residuals of equation 7.2. . . 59

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0.1 Acknowledgments

At last finished writing the thesis, time to start with the preface and acknowledgments, as it ended up. I first want to recollect parts of the process leading forward to the final work. First of all the data set that I have been working on, was eight months late. The huge amount of information (480MB) was badly documented, the data had to be massaged and information combined from different files, before ending up with an acceptable data material. That was some of the bad experience.

Of the good experience is all the people that have been willing to help me forward with their knowledge and support and made it all possible. I want to thank Sverre Holm as my supervisor in Signal Pro- cessing, without his advise and knowledge the thesis would never have been where it is today. Ole Christian Lingjærde for his supervision and knowledge in statistics, always willing to talk and share his knowledge.

To Kværner for supplying the project and sharing information and data, Tone Schanke as head of the project at Kværner for always being helpful and supplying the necessary information.

My Family for always being there. Katrine for her support through the ﬁnal stages. Ann Philips for reading correction on my (Nor)English.

USIT (Universitetets senter for Informasjons teknologi) for letting me use their oﬃce space and equipment as a part time employee. To RF (Realistforeningen): Skål kamerater... And everybody standing me close, for being my good friends.

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0.2 Abstract

In this thesis, signal processing is performed on the output from the pulsed laser photoacoustic instrument monitoring crude oil in water.

The instrument is constructed to perform inline monitoring of produced water in the pipeline during production. It is highly sensitive and testing was performed with hydrocarbons in water with concentrations in the range 0 - 1200 parts per million (ppm).

The thesis discusses the basic theory behind photoacoustic, and the construction of the instrument. Data material acquired during testing of the instrument is explored to improve the accuracy of the instrument.

The oil concentration is known to be aﬀected by the following variables: photoacoustic response, temperature, salinity and pressure. These variables are analysed with statistical regression to show the instrument’s ability to calibrate a speciﬁc compound crude oil.

Different methods of signal processing are used to enhance the result. When filtering, linear phase is necessary to avoid amplitude distor- tion of the peaks in the signal. This led to the use of a technique called spectral moments, a method that works directly on the Fourier spectrum and is insensible to phase. Statistics show that the spectral moments are able to enhance the result when equating the oil concentration. A new method for equating the oil concentration by filtering and arithmetic mean of the signal is discussed. The method is linked to the spectral moments with Parseval’s theorem, it is easy to implement and statistics show good performance.

The thesis points out that the problems with a fouled instrument window must be solved to get the accuracy of the instrument down to the expected±100ppm.

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Chapter 1

Introduction

1.1 Background

The production of oil and gas is one of the largest industries in the world today. Crude oil is reﬁned and used in combustion engines and is the key component in products made of plastic.

In the North Sea, oil platforms are a necessity in production of oil.

Research has made it possible to retrieve oil at even greater sea depths.

The construction of oil platforms is expensive, and can render a poten- tial oilﬁeld unproﬁtable.

During the production of oil and gas signiﬁcant amounts of water are pumped up to the installation. The oil, gas and water solution is then put in a separation tank, and each part is extracted. The oil and gas is transported to shore, and the water is discharged to sea. This discharge is called produced water and contains approximately 1-2 % of hydrocarbons along with various levels of dissolved hydrocarbons, sediments, heavy metals, dissolved gases such as carbon dioxide and a number of chemicals used during the production process.

The monitoring of crude oil is important and necessary because environmental legislation demands such monitoring, but also because of the importance of keeping control over the production process. Meth- ods already exists for oil in water monitoring. These methods require the aid of a person to withdraw a sample from the production line to be analysed further. The traditional method uses chemicals and solvents to analyse the sample, it takes time and is expensive both in labor and chemicals. Other methods involve looking at the absorption from different light spectra. All these methods have individual drawbacks and disadvantages, and are dependent on analysing a sample in the laboratory. Today there are no good methods for inline oil in water monitoring.

The main object is to ﬁnd a method that satisﬁes the demands of both environmental laws and the oil industry.

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2 Introduction

New technology makes it possible to move the oil production to platforms installed at the sea ﬂoor, thus cutting the construction expenses, making it possible to develop new oilﬁelds. This type of platform is called a downhole installation. The name originates from the fact that the oil separation process is performed in the bore hole. In a downhole installation every process must be fully automated. These are some of the reasons the oil industry is on the lookout for a stable system that is able to do inline monitoring of produced water.

A promising and suitable instrument, using photoacoustic, was under development at Heriot-Watt University in Edinburgh. Kværner joined in the research and further development of a prototype at the request of Norsk Hydro. The prototype researched in this thesis is constructed to accommodate specifications needed to monitor produced water from a topside oil-platform, but a downhole version is also considered. At the request of Kværner the section of signal processing at the university of Oslo was asked to contribute in the project. After a two week period of testing in the first quarter of 2000, data was collected to analyse the performance of the prototype Oil In Water Monitor (OIWM). The main issue was to find an equation that can be used to compute the oil concentration from the parameters influencing the instrument response.

The result of this analysis is found in chapter 5. The rest of the thesis is focusing on the signal produced from the instrument. New methods of analysing the signal are proposed in chapter 6. The Fourier spectrum illustrates the important components in the signal. Different types of filters are constructed and tested. Another method that is used with suc- cess is the spectral moments. The different methods are then analysed with statistics in chapter 7 to find the best one. The result is a simple to implement and efficient method that is a combination of filtering and the spectral moments.

1.2 Marine Pollution

The problem of pollution has become an issue aﬀecting all humanity.

As a result of environmental conferences and international agreements certain legislation exists around the legal amount of pollution that can be discharged into our environment. Such legislation must be followed by the oil producers in the North sea. The process of producing oil gives a certain amount of pollution, either accidentally or intentionally. An accident may occur during production that can result in high oil spills, and can have severe impact on the environment in both the short and long term. In the production of oil there is also some discharge into the sea that is accounted as normal. Such discharge can result in long term eﬀects if not kept at a minimum and under control. Figure 1.1 shows the

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1.2. Marine Pollution 3

Figure 1.1: Correlation between discharge of oil and produced water [1].

discharge in the Norwegian sector in the years 1993 to 1999 [1]. The discharge of oil or oily water into the sea is prohibited by an international agreement called the Prevention of Oil Pollution Act 1971 [2]. There are certain exemptions given to this act for oil producing installations. The Oslo and Paris Commissions (OSPARCOM) has suggested the value for legal discharge set at 40 milligrams of oil per liter (mg/l) should not be exceeded over a monthly average. This control is done with lab measurement twice daily, no more than 4% of the samples each month must exceed 100mg/l. The term more widely used to describe oil content is part per million (ppm). Due to the relative density of oil and water the concentration will be approximately the same in both units. A topside installation can per legislation discharge produced water that has an oil content up to 100 ppm, compared to 200 ppm for downhole installations. The reason for this is that the discharge is happening further away from the sea surface, and has less impact on the environment.

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Chapter 2

History and theory of photoacoustics

2.1 History of photoacoustics

The presented history of photoacoustic is an extract from a book written by Rosencwaig [4] in 1980. The photoacoustic effect goes back to Alexan- der Graham Bell who first observed it while working on a communication device in 1880. He was working on a way to transmit sound without any cables. By intensity modulating a light beam he used a selenium cell to pick up the change in light intensity and converted it to audible signals. He made the discovery that the signal could be attained directly without the electrical equipment. If the light was rapidly interrupted and focused on a solid (selenium), an audible signal could be picked up through a hearing tube. These discoveries were then further investigated and published in 1881. Similar experiments on gases were performed by John Tyndall and Wilhelm Röntgen in 1881 after hearing about Bell’s discovery. Then the field lay dormant for 50 years until the discovery of the microphone made it possible to enhance the measurements. In 1938 Viengrov at the State Optical Institute of Leningrad used the method to study infrared absorption in gases and the gas content in gas mixtures.

Pfund developed a gas analyser in 1939 in use at John Hopkins Hospital in Baltimore to measureCO andCO2. Luft developed a commercial gas analyser which became available in 1946. The interest in photoacoustics grew in this period and it was only used to monitor gases. When the infrared spectrometer was invented, a more accurate method for monitoring gases existed so the field again lay dormant until 1970. It was the invention of the laser that gave the field of photoacoustics new possibilities. The effect was earlier called optoacoustic, but the name was changed to photoacoustic by Rosencwaig. This was to avoid confu- sion with the term acousto-optic effect which refers to another physical

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2.2. The physics behind photoacoustic 5

phenomenon. The coherent, high optical energy in lasers was highly efficient for use in photoacoustics. The first use of photoacoustics with lasers was reported by White on solids in 1963, and Askar’yan et al. on liquids in 1964. New theories and understanding of photoacoustics was developed. This has led to quite an active field, today photoacoustics is being used in many different applications, and research in new areas is being done.

Photoacoustics is used in medical applications to measure glucose in samples [5], phantoms and human blood, it has been researched as a way to detect cancer, determination of melanin in human hair. Much research is performed in the detection of oil content in water. Most of this research has taken place at Heriot-Watt University in Edinburgh Scotland. Some Articles and Three Ph.Ds have been written about the subject; Hodgson in 1994[6], Freeborn in 1997 [3], and Hannigan [7] in 1999.

2.2 The physics behind photoacoustic

Much theory has been developed to explain the phenomenon of photoacoustics. This section presents the fundamental elements in understanding the physic behind photoacoustics. High energy light is used, usually a laser, ﬁring into the test sample. When the laser hits the sample, some of the energy is absorbed by the molecules in the media resulting in a region of higher temperature. The rise in temperature will generate an expanding region and a pressure wave will propagate away from the source. This decaying pressure wave is then picked up by a piezoelectric ceramic transducer. The phenomenon is contributed to the fact that the molecules in the sample being monitored have a higher response to light than the surrounding media. The process is pictured in ﬁgure 2.1.

The rest of the thesis will focus on photoacoustic propagation of hydrocarbons in water with special focus on the generated signal. There is one important formula that governs photoacoustics, and various au- thors have presented versions of this formula in their work. They have introduced particular assumptions and approximations which are relev- ant for their speciﬁcations. The formula used here is presented in the work of Freeborn [3, 8]. His formula looks especially at photoacoustics used in water, and the special cases that arise in this application.

P (r )=kE₀ αβv¹² cpt

3 e2r¹²

(2.1)

In formula 2.1 the magnitude of the pressure pulse is expressed byP (r )

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6 History and theory of photoacoustics

Figure 2.1: Principle of Photoacoustic propagation [18].

at a distancer from the optic axis, where E₀ is the incident optical energy,βis the volumetric thermal expansion coefficient,v is the acoustic velocity, cp is specific heat at constant pressure,kis a system constant andt_e is the effective time parameter. It relates the optical pulse width tp and the acoustic transit time across the optical beam radius R, and ta = R/v , such that te = (t_p² +t_a²)^1/2 . The formula 2.1 assumes relative weak absorption, as applies to water. That means that the optical absorption coefficient α and the optical path length l will define the magnitude of the acoustic region, and assumeαl 1. In the weak absorption case the geometry of the acoustic signal can be regarded as cylindrical. It is also assumed that there is no thermal diffusion in the sample. The photoacoustic response, that makes it possible to monitor oil level in water relates to the high values ofβand smaller values ofc_p in crude oil compared to water.

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Chapter 3

Instrumentation and test site environment

3.1 Instrumentation

The Oil In Water Monitor (OIWM) that is analysed in this thesis is a prototype developed by Kværner Oilﬁeld Products together with Heriot-Watt University at the request of Norsk Hydro. The goal of the tests was to collect enough information and show that it is possible to construct a OIWM that can be put into production. Kværner has done the data col- lection and ﬁeld-testing [12] of the prototype instrument at the Norsk Hydro research center in Porsgrunn. The OIWM prototype can be split into three main parts, the Head Unit, the Control Unit , and the User Interface PC.

3.1.1 Head Unit

The head unit is a housing that is capable of withstanding 100 bar of pipeline pressure. Light from the laser diode drivers in the control unit is transferred by a fiber-optic cable. The laser then enters the sample through a sapphire window. Below the sapphire window a half moon shape is protruding, this is the location of the Piezoelectric ceramic Transducer (PZT) used for acoustic detection. The positioning is illus- trated in figure 3.1. The PZT detects photoacoustic pressure pulses in the range of 0−0,5P aat ambient temperatures. The PZT output voltage is in the area 5−10uV. The PZT is glued to a lead backing that dampen vibrations, but still keeps a strong photoacoustic response. Within 5 cm of the PZT a pre-amplifier card is placed to reduce attenuation. This amplifies the signal 62dB, and result in a output signal of 40mV to 400mV. A pressure transducer is mounted on a flange, and is expected to monitor pressure levels less than 100 bar, in a process environment

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8 Instrumentation and test site environment

Figure 3.1: Front view of the head unit [3].

up to 100^◦C. Part of the light is directed to an energy monitor with a beamsplitter, this way the laser energy and eventually any damage on the ﬁber-optic cable can be detected. The head unit includes a temperature sensor fastened to the housing with temperature conductive epoxy, for monitoring of pipeline temperature.

3.1.2 Control Unit

The Control Unit is connected to the head unit by an armored cable, through this cable laser pulses and information from the sensors in the head unit passes. The control unit contains two pulsed diode lasers, one with wavelength 905nm and another with wavelength 1550nm, firing sequentially. The light is transferred to the head unit through a fiber- optic cable. Due to losses in the system (couplers and fiber) only 30%

of the energy is emitted into the sample ﬂuid. The control unit contains the power supply for laser and electronics, together with PC-hardware running DOS. It also contains the fast analog-to-digital (A/D) card with 8 bit resolution that samples the signal and stores it on local disk. A keyboard and monitor allows direct control of the data acquisition and logging. The Control Unit operates independently, some of the data is sent via a serial link to the user interface PC.

3.1.3 User Interface PC

The User Interface PC is a standard PC running windows 95. Specialized software is run and displays output from sensors. It indicates system status and display alarms when values reach a preset threshold.

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3.2. Oil in water monitor test site environment 9

3.2 Oil in water monitor test site environment

The OIWM was tested at Hydro research center in Porsgrunn during the ﬁrst quarter of 2000. One of the goals was to ﬁnd an equation that ex- presses the oil concentration with respect to the variables salinity, pressure and temperature, and to get an idea of the error in such a formula.

The values tested against reﬂect the values expected to be found in produced water. The test procedure was as follows; In the ﬁrst test period the aim was to get data of the oil concentration versus temperature and photoacoustic response. Three temperatures were used as basis , 30^◦C, 40^◦Cand 50^◦C. At the same time the oil concentration was varied along the values 0ppm, 300ppm, 600 ppm, 900ppm, and 1200 ppm. In the last test period salinity was tested with various temperatures and oil concentrations. Pressure was also tested in the range of 1−30bar. No change in photoacoustic response was observed [18] as a function of pressure in this range and has therefore been omitted in further analysis of the dataset. The test variables and their values are displayed in table 3.1.

Test variable Range Unit

Oil concentration 0, 300, 600, 900, 1200 ppm Temperature 30, 35, 40, 45 ^◦C

Pressure 1, 10, 20, 30 bar

Salinity 30 ,38, 45, 48 g/l

Table 3.1: Test parameters.

The instrument head was inserted into an one inch pipeline, normal sea water with salinity 30g/lwas pumped through the system introdu- cing oil and additional salt when necessary. A pressure valve was used to regulate the pressure. An oil dispenser was used to set the oil concentration. Figure 3.2 shows the OIWM installed in the water rig at Hydro’s research center at Porsgrunn.

During testing one or two samples were retrieved from the pipeline during a datapoint, and the oil concentration was measured with infrared spectrophotometric method [10] (IR) at a later time. When two samples existed the mean of the two values was used. The IR-method follows the Norwegian standard that is currently in use at oil installations. The oil concentration is not known exactly, because both the oil dispenser and the IR-method include measure errors, the correlation between the two methods is shown in ﬁgure 3.3. At this point there is uncertainty as to which of the two methods gives the best value. Be- cause of this uncertainty both the IR-method and the dispenser values are used when trying to ﬁnd models expressing the oil concentration,

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Figure 3.2: OIWM installed in the water rig at the Hydro’s Research Cen- ter in Porsgrunn.

Figure 3.3: Plot of the IR-method versus the dispenser oil concentration.

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3.3. Signal description 11

each is presented with separate analysis. Any error in the reference method will be inherited by the models constructed from these measurements. Not all the data points were monitored with the IR-method, so less data is available in this case. The ﬁrst week of testing Brage oil was used, this is a relatively thick oil and stuck easily to the OIWM window, resulting in measure errors in the form of heightened response. No results were obtained with this oil. Instead the thinner Visund oil was used in the rest of the period, giving better results, but still contributing to some problems. During a one day run the OIWM acted stably, but after running for several days the sapphire window became fouled and resulted in an increased response. The window was cleaned to get the response back to base level. This procedure was performed three times during the Porsgrunn testing.

3.3 Signal description

The control unit receives the amplified PZT signal from the instrument head. The signal is processed with a fast Analog to Digital card (A/D), with a sampling frequency at 100MHz, and with a precision of 8 bit. The card acquires 1000 samples at intervals of 10 nano seconds. In earlier experiments amplitude analysis has been used to find the photoacoustic response. This value is found by looking at the difference between the highest and lowest peak of the signal within a specified window. This value is called the peak-to-peak value, a typical signal with the peak-to- peak value marked off is shown in figure 3.4.

3.3.1 Signal averaging

The signal generated from one pulse of the laser contains much noise, and it is nearly impossible to distinguish any special features. The noise is white, and comes from induction in cables, electronics and physical vibrations picked up by the transducer. To enhance the signal-noise ratio an averaging of signals is performed. Earlier testing [13] show that an average over 1500 pulses is reasonable for the 905nm wavelength, and 500 pulses for the 1550nm wavelength. It was believed that the accuracy could be further improved so the prototype uses averaging over 1500 pulses times 10. The instrument averages in steps of 1500 pulses, each time ﬁnding the peak-to-peak value of this average. The ten last peak-to-peak values are averaged and the value is written to the log ﬁle.

This value is called a datapoint and consists of a total average of 15000 pulses. The waveform signal, as shown in ﬁgure 3.4, consists of the average of the last 1500 pulses of the ten and is also written to log ﬁle at this time. This means that the peak-to-peak value found from the waveform

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0 1 2 3 4 5 6 7 8 9 10

−30

−20

−10 0 10 20 30

Time (ms)

Photoacoustic response

Peak−to−peak value

Propagation delay

Figure 3.4: Typical photoacoustic signal from the OIWM.

signal consists of an average of 1500 pulses compared to a datapoint consisting of 15000 pulses. Because of this averaging process, the peak- to-peak value from a datapoint will have a better signal-noise ratio value compared to the peak-to-peak found from the waveform signal.

3.3.2 Physical dependencies

Earlier laboratory experiments [3] have shown that the photoacoustic response is connected to the level of salinity and temperature. This relationship follows from equation 2.1. The key physical parameters are the thermal expansion coefficient, β, specific heat capacity, cp, and acoustic velocity,v, all vary with both temperature and salinity. It is also known that equations expressing acoustic velocity in water are affected by pressure. Pressure was tested again at the Hydro research center at Porsgrunn to see if it affected the photoacoustic response, but no dependencies were visible in the range up to 30bar [18].

Acoustic propagation delay

The time between delivery of the optical pulse and the detection of an acoustic signal is called the acoustic propagation delay. This delay,t_pk, is determined by the acoustic speed in water,vand distance to the transducer, r. In addition comes a system constant that relates to the delay

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3.3. Signal description 13

in system electronics, tsys. The relation is expressed in 3.1, and is displayed in ﬁgure 3.4.

t_pk= r

v +tsys (3.1)

The acoustic velocity in water is well documented and is known to only be related to the three physical quantities, temperature, salinity and pressure [22]. No other physical properties have been found to aﬀect the velocity of sound in sea water. It is reasonable to believe that it is possible to ﬁnd the salinity from the above mentioned parameters, and use this salinity when equating the oil concentration. This issue is researched further in section 5.6.

The eﬀect of optical energy on signal response

When calculating the photoacoustic response, the optical energy enter- ing the sample has signiﬁcant eﬀect. In laboratory tests the peak-to-peak signal was found to be directly proportional with energy [3]. The energy monitor in the head unit picks up the energy in incident optical light, this value is used to normalize the signal. In this way variations in optical energy can be largely disregarded. All values used in analyses in this thesis use the normalized signal.

The eﬀect of crude oil on signal response

Crude oil is made up from a long range of different hydrocarbons, each type comes in different quantities, depending on the oil reservoir. The number of carbon atoms in a hydrocarbon molecule describe the mo- lecular size. High quantities of large molecules in a sample, result in high photoacoustic response. The photoacoustic response of crude oil was tested against different wavelengths of light [3]. The difference in response between crude oil and salt water shows better characteristics at the 905 nm wavelength than the 1550 nm wavelength. The 1550 nm wavelength was choosed for the low response to crude oil to possible detect other effects. It was primary added to be researched as a base level detector that could be used to remove the effect of the oil film on the instrument window. This was not seen as an important issue by Kværner and is the reason the 905nm wavelength is the only light source used in further analyses.

Other test-parameters

If the instrument is to work in production all consideration must be accounted for. The performance when the produced water contain particles, chemicals, and gas, and the reaction towards ﬂow rate and droplet size

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encountered in the pipeline. These were important questions to answer before starting development of the prototype.

These parameters were tested at Norsk Hydro Research Center in Porsgrunn in the last quarter of 1998 [11]. The droplet size was tested in the range 20−30µmand no changes was recorded in the photoacoustic response. Theoretically the ﬂow rate is not expected to change the photoacoustic response. The measurement can be seen as a snapshot within a period of 5µs, in this period the sample will have moved 10µm which is small compared to the acoustic wavelength of around 1.5mm and the travel length to the PZT. The instrument insensibility to ﬂow was proved in these tests.

To test the inﬂuence of gas, nitrogen was injected into the pipeline.

The result shows a decrease in signal with increasing amount of gas.

This can be the cause of scattering of the laser, or because nitrogen ab- sorbs the wavelength. When the amount of nitrogen was kept constant, the photoacoustic response showed a constant linear trend, no further tests have been performed on this issue.

More tests were run with sand and Bentonite. Sand does not seem to change the result, while Bentonite showed a decrease in the photoacoustic response, few test points where run so it is hard to do draw a firm conclusion. Chemicals were run with methanol up to 3 %. Dissolved hydrocarbons were seen as one of the big challenges, but little influence on the photoacoustic response was recorded. The conclusion after these tests was that the important factors that influence the photoacoustic response are pressure, salinity and temperature.

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Chapter 4

Methods of analysis of the photoacoustic signal

4.1 Spectral moments

The spectral moments is a method that retrieves information directly from the Fourier spectrum. It is advantageous when we wish to extract special features in the signal and is not dependent on keeping the original signal itself. One of the advantages is the insensitivity to changes in the phase of the signal. Spectral moments use the power of the Fourier spectrum when making calculations. Filters that distort the phase might ruin the signal, but the spectral components will stay untouched. The spectral moments are a method usually used in image processing [16], the central moments are derived from the spectral moments and work as a transformation that can mirror image, rotate and resize an image.

The spectral moments can also be viewed as a way to use statistics to analyse the power spectrum. The diﬀerent moments have a statistical interpretation that will be described below.

The one dimensional spectral moment [17], m_n, of sizen of a continuous power spectrumS(f )is deﬁned in 4.1.

mn= _∞

0 fⁿS(f )df (4.1) When working on a discrete-time signal the spectral moment is deﬁned in equation 4.2. The length of the power spectrum isN, and the sampling interval is∆t.

mn= 2 N

(N/2)−1 l=0

S(l)(l/N∆t)ⁿ (4.2) We wish to limit our computations to a band of frequencies, this will correspond to a ﬁltering process where the frequencies of interest are

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16 Methods of analysis of the photoacoustic signal

extracted. In the continuous case we set the frequency limits of the integral fromc1 toc2, and get equation 4 .3

mn= c₂

c1

fⁿS(f )df (4.3)

In the discrete case the spectral estimate is done in the frequency range c1 toc2 of length I, whereI =c2−c1. The sampling interval is still∆t.

The result is equation 4.4 mn= 1

I

c₂

l=c1

S(l)(l/N∆t)ⁿ (4.4) The zero order moment will in the discrete case look like equation 4.5.

m0 = 1 I

c₂

l=c₁

S(l) (4.5)

The zero-order spectral moment is basicly the average of the values in the interval c1 to c2, and is proportional to the mean energy in that interval. If we use equation 4.2 to take the zero order spectral moment over the whole signal, it can be estimated directly from the variance from the time series itself. When only working on parts of the spectrum it is necessary to normalize the spectrum withm₀ before taking the higher spectral moments. This will level out the effect of the size of the values of higher order moments, and make it possible to extract information about the shape of the spectrum. The first order spectral moments in the continuous case used on part of the spectrum is defined in equation 4.6

m1= 1 m0

c2 c1

fⁿS(f )df (4.6) The discrete ﬁrst order moment will then look like equation 4.7

m₁= 1 Im0

c₂

l=c₁

S(l)(l/N∆t)ⁿ (4.7) The ﬁrst order spectral moment is interpreted as the mean frequency of the signal in the intervalI. To get information about the higher order moments one needs to deﬁne the central moments 4.8.

Mn= 1 m0

_c₂

c₁(f −f )¯ ⁿS(f )df (4.8) The mean frequency, ¯f is found from the 1. order moment. The discrete version of this formula is given by equation 4.9

Mn= 1 Im0

c₂

l=c1

S(l)( l

N∆t −¯l)ⁿ (4.9)

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4.2. Linear regression analysis 17

Where the mean frequency ¯lis found from the discrete 1. order moment.

The second order central moment can be interpreted as the variance of the power spectrum, or the squared bandwidth. The third order is the skewness, and the fourth order is the kurtosis. The central moments can be easily found from the moments.

4.2 Linear regression analysis

A method that tries to ﬁt an equation to a set of data is called regression. Among the various methods of regression, linear regression is the simplest, and the most widely used. Descriptions of the method are found in almost any elementary textbook in statistics [15]. In two dimensions a line may be described in algebraic terms on the form 4.10.

y=α0+α1x (4.10)

In the real world, observations usually do not ﬁt an equation like this, even if the true relationship is linear. A term expressing the noise in each observation must be added. The model then becomes

yi=α0+α1xi+*i (4.11) where the subscript i in the equation stands for observation number i.

The term*_iis the noise in the observation. In general we may have more than one independent variable, and a multiple regression model such as y_i=α0+α1x_i1+α2x_i2+ · · · +α_kx_ik+*_i (4.12) may be considered. This model hasi=1, . . . , nobservations andk≤1 independent variables. The predicted values ˆy_iare expressed with

yˆi=αˆ0+αˆ1xi1+αˆ2xi2+ · · · +αˆkxik (4.13) where ˆα0, . . . ,αˆ_k is the estimate ofα0, . . . , α_k in equation 4.12. The residual error can then be deﬁned as

e_i=y_i−yˆ_i (4.14)

where y_i is the observation, and ˆy_i is the predicted value found using formula 4.13. Equation 4.12 may be written in matrix notation as





 y1

y₂ ... yn





=







1 x₁₁ x₁₂ · · · x_1k 1 x21 x22 · · · x2k

... ... ... . .. ... 1 xn1 xn2 · · · x_nk











 α₀ α1

... αk





+







*₁

*2

...

*n





 (4.15)

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or simply as

y=Xα+* (4.16)

where*,y∈IRⁿ,X∈IR^n,k+1, andα∈IR^k+1. The goal is to ﬁnd a method to estimate the parametersαj. One way to obtain an estimate for αis by minimizing the square sum of the error.

S= n i=1

*²_i = n i=1

(y_i−α₀−α₁x_i1−α₂x_i2− · · · −α_kx_ik)²

=(y−Xα)(y−Xα)=yy−2α(Xy)+α(XX)α (4.17) In order to ﬁnd the minimizer, we use matrix diﬀerentiation on equation 4.17 and get equation 4.18

δS

δα = −2Xy+2XXα (4.18)

Setting equation 4.18 equal to zero, we see that the least squares estimate satisﬁes

(XX)α=Xy (4.19)

IfXXis a non-singular matrix, equation 4.19 has the unique solution

α=(XX)⁻¹Xy (4.20)

The vectorαthen gives the minimum of S in equation 4.17.

4.3 Variable selection

Frequently we start with a long list of independent variables that we sus- pect have some eﬀect on the dependent variable, but for various reasons we want to reduce the list. To test the eﬀect of thejth term the following hypothesis may be considered:

H0:aj =0 ver sus H1:aj=0

The decision we want to make is whether we should accept or reject the hypothesis that variable a_j is equal to zero. Using Student’s t-test we consider the test statistics

t= aˆ_j

se(aˆ_j) (4.21)

Which, under the assumption of normal errors, follows a Student’s t- distribution with one degree of freedom under the hypothesisH0 where se is the standard error of the independent variable. The ˆa_j is the estimated value of the variableaj. The p-value is deﬁned below

p=pr ob(T >|ˆt_obs|) (4.22)

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4.4. Speed of sound in water 19

where T follows a Student’s t-distribution with one degree of freedom.

The smaller the p-value, the more significant is the corresponding term in the model. The term ˆtobsindicates if a observed value in the model lie within or outside the estimated value in interval(−ˆt,t)ˆ and ˆt∼t. Under normal assumptions we will conclude that a parameter is significantly different from zero if the p-value is less than, lets say 0.05. Another method used to decide on variables and comparing models is Mallow’s C_p, and is defined as:

Cp = RSS

s² −(n−2p) (4.23)

Here RSS is the Residual Sum of Squares and is deﬁned asn

i=1e²where eis the residual from equation 4.14. The term p is the number of independent variables, n is number of observations, and s² is an unbiased estimator ofσ². The terms² is deﬁned as

s²= n i=1

e²_i

n−k−1 (4.24)

where the term (n-k-1) is the degrees of freedom, where k is number of independent variables, and n is the number of observation.

The purpose of variable selection procedures is to select or help select from the total number of candidate variables to a smaller subset.

A number of procedures can be performed, selecting the best candidate with eithers²,Cp or possible other types of statistic. In many situations there is rarely one obvious best equation, and the near winners are almost as good. One approach when searching for the best model is to in- spect all possible subsets. Another method that consume less resources is a stepwise selection, most statistical programs have procedures for removing the less important variables one by one. The method of stepwise selection often removes important variables from a model. It will often be necessary to do manual work and it is advised that the analyst use his or her intuition.

4.4 Speed of sound in water

The equations used to measure the speed of sound in water include the three variables temperature, salinity, and pressure. No other physical quantities affecting the value are found. The cross correlation between the three terms appears to be complicated. Tables have been widely in use to find the relation, but around 1970 equations are able to express the value more accurately. Del Grosso [23] gives an equation containing 19 terms each to 12 significant figures in the powers and cross products

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of the three variables. Lowett [24] presents a less cumbersome equation, but still values of produced water are far outside the values used in his dataset. H. Medwin has a simpliﬁed equation with a larger error with limits in an interesting range [22] as seen in equation 4.25

c= 1449.2+4.6T−5.5×10⁻²T² 0≤T ≤35^◦ +2.9×10⁻⁴T³+(1.34−10⁻²T )(S−35) 0≤S≤45 +1.6×10⁻²D 0≤D≤1000

(4.25)

T is the Temperature in Celsius, S is salinity in parts per thousand, D is the depth in meters, and c is the velocity of sound in water. The depth variable in equation 4.25 corresponds to the pressure variable used in the OIWM analysis.

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Chapter 5

OIWM Estimation of Parameters for Linear Models

5.1 Introduction

The instrument has been through a series of testing at the Norsk Hy- dro research center in Porsgrunn, the testing was performed in the ﬁrst quarter of 2000. Kværner performed the ﬁnal preparation of the data.

The purpose of this chapter is to propose some linear models produced from the data that can be used to predict the oil concentration.

The photoacoustic response used in this analysis is the normalized peak to peak value. This is the value that is automatically equated by the OIWM during testing. To represent the oil concentration we use both the oil dispenser and the IR-measurements. We want to model the two on behalf of photoacoustic response, temperature and salinity. The test matrix and other circumstances around the testing are described in section 3.2.

5.2 The test setup

Prior to the Porsgrunn tests an older version of the instrument had shown good results in laboratory experiments both at Heriot-Watt university, and in ﬁeld experiments. A strong linear relationship was proved between the photoacoustic response and the oil concentration. Lin- ear dependencies towards temperature, salinity and pressure were also found. The interaction between each and all variables was not checked at this stage. The instrument used in earlier testing was replaced with a new prototype with enhanced electronics. At the request of Kværner a project to ﬁnd an equation to express the oil concentration as a function of salinity, pressure, and temperature was assigned to the section

21

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22 OIWM Estimation of Parameters for Linear Models

Temperature

30 35 40 45 50 55 60

506070

o ppm0

o ppm300 o ppm600 o ppm900 o ppm1200

Figure 5.1: All data, using oil concentration from the oil dispenser.

of signal processing at the University of Oslo. Because of the linearity observed in prior experiments, it was assumed that a linear model would be suﬃcient to ﬁnd the oil concentration. It was still a possibility of strong interaction between variables that would make a linear model inappropriate.

First let us have a look at the relationship between photoacoustic response and the temperature. Figure 5.1 is a plot of all the data acquired during testing, using the values from the oil dispenser for oil concentration. The temperature varies from 30^◦C to 60^◦C. The oil concentration is varying in the area from 0 to 1200ppm in steps of 300 ppm. The plot can be viewed as a three-dimensional figure, where the third axis is perpendicular to the paper. The oil concentration is colour coded to il- lustrate this effect. The plot exhibits the expected linear trend observed in earlier experiments. The effects of salinity and pressure are not accounted for in these plots. The results at higher oil concentrations might indicate that a nonlinear model will be more appropriate, however this is difficult to decide on the basis of the current data.

Figure 5.2 shows another way of presenting the data. The figure displays the oil concentration using the IR-method versus the photoacoustic signal for 30^◦C, 4 0^◦C and 50^◦C. The oil concentration from the IR-method varies, so the temperature is the fixed parameter instead of the oil concentration. The plot in figure 5.2 does not consider the temperature variations of 2−3^◦C at the different temperature levels. This

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5.2. The test setup 23

Oil concentration, IR-method (ppm)

50 55 60 65 70 75

020040060080010001200

o temp30 o temp40 o temp50

Figure 5.2: All data, using oil concentration from the IR-method.

temperature change gives at first glance at figure 5.1 a change of about 2-3 in photoacoustic response. This will influence the placement of the data. Samples of the IR-method were collected once or twice during a test point. When more than one sample were collected, the mean of the samples were used in further analysis to represent the oil concentration.

The variations in the samples were often more than the ±10% that the IR-method is known to exhibit [25]. This can be the result of changes in values delivered by the oil dispenser, background noise or measurement errors. Little information exists about the general stability of the oil dispenser. The plot in figure 5.2 contains less data than figure 5.1, because the oil concentration was not checked with the IR-method at every run. In figure 5.2 the photoacoustic response relates to the IR- method at different temperatures. The IR-measurement is clustered in horizontal lines, where each line corresponds to one datapoint. This also illustrates the variations in the photoacoustic response within each datapoint. These variations can be due to change in temperature or dif- ferences in oil concentration delivered by the oil dispenser during a run.

The plot shows that the IR-method relates linearly to the photoacoustic response within the three ﬁxed temperatures displayed. From ﬁgure 5.1, and 5.2 it is believed that a linear model will be appropriate between the oil concentration and the temperature in further analyses. It is also believed that temperature is the variable that has the largest impact on the photoacoustic response.

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5.3 Linearity in data

We start the analysis by exploring the linearity of a single level of oil concentration, ﬁrst working on the data acquired from the oil dispenser.

A separate analysis is performed on the ﬁve diﬀerent oil concentrations.

The level of salinity is kept constant at 30g/l; this is the natural salinity of normal sea water. This means that the salinity predictor is removed and we are working on a smaller dataset than shown in figure 5.1. To ensure that the oil concentration used in the regression is independent variables we use the mean of the values within each datapoint. It is the true oil concentration we want to model, but we use the dispenser oil and IR-measurements to represent the value. It is this value that will be used in the analyses for the rest of this chapter. The reason is that the oil concentration, both the IR-method and the oil dispenser, is the same within a datapoint. This behavior can affect the output values of standard deviation and the p-value. Each datapoint is therefore counted as one observation. The total dataset is then reduced from 1607 observations to 84observations. At this point it is still possible to visualise the data graphically. Linear regression is used to find a linear fit for each of the five oil concentrations. The five regression lines are displayed together with the data in figure 5.3. The figure shows that the photoacoustic response increases with increasing oil concentration.

This is the same result that is observed in earlier experiments. At the higher oil concentrations inconsistency occurs, the oil concentration of 1200 ppm intersects the line of 900 ppm. This plot can help to un- derstand and check the quality of the data. It is believed that many of the problems at higher oil concentrations arise from the fouling of the instrument window. Over time an oil layer builds up on the instrument window, resulting in a heightened photoacoustic response. Other sources of error can be bookkeeping or problems with the oil dispenser at higher oil concentrations. In the last case a better ﬁt can be achieved with the IR-method. It is therefore interesting to look at the linearity of the data monitored with the IR-method. This analysis includes only the salinities of normal seawater, data with values outside this range is removed. Two datapoints were found to have salinity higher than normal seawater, and were removed. The oil dispenser method had eighteen such datapoints in comparison. Only two salinity values with concentration higher than seawater make a poor basis when trying to construct models that consider salinity as a predictor. The size of the dataset is more than halved because fewer datapoints was monitored with the IR- method. The method using the oil concentration from the dispenser has 66 observations in comparison to the IR-method with 31 observations.

Many observations are usually advantageous when doing statistical analysis, helping towards a better and more accurate result. In this analysis

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5.4. Estimation in three dimensions 25

Temperature

35 40 45 50 55 60

5055606570

ppm0 ppm300

ppm600 ppm900 ppm1200

ppm0 ppm300 ppm600 ppm900 ppm1200

Figure 5.3: Data showing photoacoustic response versus temperature with corresponding regression lines.

we wish to display the relation between the photoacoustic response and oil concentration from the IR-method. Each of the three temperatures 30^◦C, 4 0^◦C, and 50^◦C is displayed in the same plot. We are using the mean of each datapoint with the same reasoning as mentioned previ- ously. The plot is displayed in figure 5.4and can be visualised as a three dimensional space with the colour-coded temperature perpendicular on the paper. Linear regression is used to fit the lines. The plot shows few available values appearing above the oil concentrations of 800ppm, and the values appearing at this level display a bad fit to the regression line.

This indicates that problems at higher oil concentrations might exist when using the IR-method.

5.4 Estimation in three dimensions

By constructing a model that includes the three most important predictors the oil concentration can be estimated as a function of photoacoustic response and temperature. Salinity is still excluded outside levels of normal seawater. First a model based on the dispenser oil concentration is investigated. The oil dispenser level is a set value with ﬁve levels, and not containing noise in the statistical sense. When constructing models based on the dispenser oil concentration we must be aware of this fact.

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Oil concentration, IR-method (ppm)

55 60 65 70

020040060080010001200

temp30 temp40 temp50

Figure 5.4: Data showing photoacoustic response versus IR-method oil concentration with corresponding regression lines.

The oil concentration is expressed with the two variables temperature and photoacoustic response, the model is displayed in equation 5.1.

oili=α0+α1pri+α2tempi+*i (5.1) Theoil_iis the true oil concentration, pr_i is the photoacoustic response of measurement numberi,temp_iis the temperature at the same instant.

Since the true oil concentration is unobserved, the oil dispenser level is used as a proxy. Therefore the term*_i incorperates observational error as well as modelling error. The error is due to diﬀerences in the set oil level and the true oil level. The model estimates the parametersα0,α1, andα2. The term*_i is the noise in the observations, even if there is no noise in the process of setting the dispenser level, there is general noise in the system. The output after performing linear regression is displayed in table 5.1. The table shows the values of the diﬀerent parameters corresponding to equation 5.1, their standard deviation and thep-value of the parameter after regression. The parameters have low standard deviation and the p-value is still close to zero. The same analysis is done with the oil concentration from the IR-method. The variableoil_i is exchanged with the variableir_i that represents the oil concentration found with the IR-method. The resulting model is shown in equation 5.2.

ir_i=α0+α1pr_i+α2temp_i+*_i (5.2)

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5.5. Estimation including salinity 27

Parameters Value Sd. p-value α₀ -2450.2 171.0 <0.001 α1 82.2 4.4 <0.001 α2 -47.1 4.0 <0.001

Table 5.1: Value of the three parameters in equation 5.1, their Standard deviation (Sd.) and thep-value.

The resulting parameters, after doing regression on the model, are shown in table 5.2. The result is similar to the values in table 5.1.

Parameters Value Sd. p-value α0 -2362.7 345.1 <0.001 α1 77.8 7.6 <0.001 α₂ -45.0 5.4 <0.001

Table 5.2: Value of the three parameters in equation 5.2, their standard deviation (Sd.) and thep-value.

The question is now, how good are these estimates?. One way to visualise the fit is by plotting the data versus the fitted values from the equations 5.1 and 5.2. The figures 5.5 and 5.6 display this relation from the equations respectively. These plots include the best fit regression line. The distance a value appears from the best fit regression line indicates how much it missed the value it was intended to fit. The monitored oil concentrations appears in horizontal lined clusters, in the case of figure 5.5 each such horizontal line represents the five oil concentrations that were used as reference value for the oil dispenser. In the case of figure 5.6 a cluster appears only at 0ppmwhere it uses the same data as the oil dispenser. The oil concentration of 0ppmwas assumed to always run pure water in the system because the oil dispenser was turned off. This level of concentration was never monitored with the IR-method.

5.5 Estimation including salinity

The last step in ﬁnding a formula that can estimate oil concentration is to include the salinity. The only changes from the equations presented in last section, is the additional salinity term. When adding the salinity term we also include more data in estimation of the predictors in the model.

oil_i=α0+α1pr_i+α2temp_i+α3sal_i+*_i (5.3)

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fitted oil concentration (ppm) values

monitored oil concentration (ppm) values

-200 0 200 400 600 800 1000 1200

020040060080010001200

Figure 5.5: Plot of regression ﬁt of the oil concentration from the oil dispenser.

fitted oil concentration (ppm) values

monitored oil concentration (ppm) values

0 200 400 600 800 1000

020040060080010001200

Figure 5.6: Plot of regression ﬁt of the oil concentration from the IR- method.

Spectral moments and linear models used for photoacoustic detection of crude oil in produced water

University of Oslo

Department of Informatics